Journal of Differential Geometry

Boundary torsion and convex caps of locally convex surfaces

Mohammad Ghomi

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We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least $4$ times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the $4$ vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical $4$ vertex theorem. The proof involves studying the arrangement of convex caps in a locally convex surface, and yields a Bose type formula for these objects.


Research of the author was supported in part by NSF Grant DMS-1308777, and Simons Collaboration Grant 279374.

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J. Differential Geom., Volume 105, Number 3 (2017), 427-486.

Received: 17 February 2015
First available in Project Euclid: 3 March 2017

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Ghomi, Mohammad. Boundary torsion and convex caps of locally convex surfaces. J. Differential Geom. 105 (2017), no. 3, 427--486. doi:10.4310/jdg/1488503004.

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